Condition numbers can also be defined for nonlinear functions, and can be computed using
calculus. The condition number varies with the point; in some cases one can use the maximum (or
supremum) condition number over the
domain of the function or domain of the question as an overall condition number, while in other cases the condition number at a particular point is of more interest.
One variable The
absolute condition number of a
differentiable function f in one variable is the
absolute value of the
derivative of the function: : \left|f'(x)\right| The
relative condition number of f as a function is \left|xf'/f\right|. Evaluated at a point x, this is : \left|\frac{xf'(x)}{f(x)}\right|=\left|\frac{(\log f)'}{(\log x)'}\right|. Note that this is the absolute value of the
elasticity of a function in economics. Most elegantly, this can be understood as (the absolute value of) the ratio of the
logarithmic derivative of f, which is (\log f)' = f'/f, and the logarithmic derivative of x, which is (\log x)' = x'/x = 1/x, yielding a ratio of xf'/f. This is because the logarithmic derivative is the
infinitesimal rate of relative change in a function: it is the derivative f' scaled by the value of f. Note that if a function has a
zero at a point, its condition number at the point is infinite, as infinitesimal changes in the input can change the output from zero to positive or negative, yielding a ratio with zero in the denominator, hence infinite relative change. More directly, given a small change \Delta x in x, the relative change in x is [(x + \Delta x) - x] / x = (\Delta x) / x, while the relative change in f(x) is [f(x + \Delta x) - f(x)] / f(x). Taking the ratio yields : \frac{[f(x + \Delta x) - f(x)] / f(x)}{(\Delta x) / x} = \frac{x}{f(x)} \frac{f(x + \Delta x) - f(x)}{(x + \Delta x) - x} = \frac{x}{f(x)} \frac{f(x + \Delta x) - f(x)}{\Delta x}. The last term is the
difference quotient (the slope of the
secant line), and taking the
limit yields the derivative. Condition numbers of common
elementary functions are particularly important in computing
significant figures and can be computed immediately from the derivative. A few important ones are given below: {\sqrt{1-x^2}\arccos(x)}
Several variables Condition numbers can be defined for any function f mapping its data from some
domain (e.g. an m-tuple of
real numbers x) into some
codomain (e.g. an n-tuple of real numbers f(x)), where both the domain and codomain are
Banach spaces. They express how sensitive that function is to small changes (or small errors) in its arguments. This is crucial in assessing the sensitivity and potential accuracy difficulties of numerous computational problems, for example,
polynomial root finding or computing
eigenvalues. The condition number of f at a point x (specifically, its
relative condition number : \lim_{\varepsilon \to 0^+} \sup_{\|\delta x\| \leq \varepsilon} \left[ \left. \frac{\left\|f(x + \delta x) - f(x)\right\|}{\|f(x)\|} \right/ \frac{\|\delta x\|}{\|x\|} \right], where \|\cdot\| is a
norm on the domain/codomain of f. If f is differentiable, this is equivalent to: : \frac{\|J(x)\|}{\|f(x) \| / \|x\|}, where denotes the
Jacobian matrix of
partial derivatives of f at x, and \|J(x)\| is the
induced norm on the matrix. == See also ==